Applications of the Laplace transform in mathematics and engineering

Abstract EN

In this paper, we define the Laplace transform as an effective mathematical tool for simplifying complex issues.

We show the sufficient conditions for the existence of the Laplace transform the paper also includes a brief overview of the inverse Laplace transform and its important properties and the inverse Laplace transform of some functions.

Then, the paper explains the mathematical and physical applications and the application of the Laplace transform to them, where we clarified the way, this transformation works and its creative idea that it transforms the differential equation into an algebraic equation that can be solved.

Using the inverse Laplace transform, the result is easily reached after converting it into ordinary differential equations or differential equations, a partial that is difficult to solve with the usual methods known and explained to obtain the time domain from its frequency domain.

The paper, also, shows the Laplace most important advantages, as it is possible to find the special solutions of the differential equations directly without the need to find the general solution in advance.

Therefore, the Laplace transform solves the inhomogeneous differential equations without the necessity to solve the corresponding homogeneous equations first.

The Laplace transform also applied in the case of continuous, discrete, complex and progressive functions.